Manasiya 1 The Babylonian Numeral System We currently live in a world where numbers are everywhere. They are in our houses, our cars, our phones and laptops, and so many more things. We may take numbers for granted, even though we use them all the time, but the numbers we use today were not the first numeral system. Many civilizations before us helped start and develop the idea of numeral systems. The Babylonians, being one of these civilizations, developed one of the oldest positional numeral system with base 10 and base 60 numbers, helped created the idea of a “zero”, and influenced many future mathematical ideas. The Babylonian civilization, one of “the oldest of the world’s civilizations, [rested in] Mesopotamia which emerged in the Tigris and Euphrates river valleys, now the southern part of modern Iraq”1, around 1700 B.C.E. The Babylonians emerged after invading the Sumerian civilization, but the Babylonians inherited much of their mathematical knowledge from their predecessor. A lot of what we know about the Babylonians comes from the clay tablets used during that time. The Babylonians would “write” or inscribe the clay tablet when the clay was soft and moist. Then they let the tablet dry in the sun to harden, so they could use the tablets again. This process allowed us to discover the Mesopotamian civilization and learn more about them. Even though “among all discovered Babylonian clay tablets, about three hundred are on mathematics”2, the tablets provide us with rich information on the Babylonian numeral system. The Babylonian number system uses a combination of base 10 and base 60 numbers. Base 10 is what we currently use with our Hindu-Arabic number system, but the Babylonians only used base 10 to represent numbers from 1 up till 59. Only two symbols created all the combinations of these numbers. To create these symbols, “cuneiform”, “which literally translates 1 Ji, Shanyu. History of Mathematics. (Lecture 1 p. 2) 2 Ji, Shanyu. History of Mathematics. (Lecture 1 p. 3) Manasiya 2 [to] ‘wedge shaped’”3, scripts were used. The number “1” was represented by a wedge down, like this: . To create 2 – 9, the downward wedge was connected by the number of wedges needed to create that number. For example, “2” was represented by , while “9” was represented by . The number “10” cuneiform was sideways , and to create the multiples of 10 up till 50, you added as many side wedges needed. Base 60 was used to create numbers greater than 59. The base 60 part of the Babylonian numeral system was “inherited from the Sumerians, who developed it around 2000 B.C”.4 To use base 60, the Babylonians developed the positional numeral system, which is very similar to our positional decimal system. The number 135 can be written in base 10 as 1*102 + 3*101 + 5, but the Babylonians wrote 135 as 2*601 + 15. To utilize the positional numeral system for 135, first read the cuneiform that follows from left to right: . The two downward wedges represent the two 60’s needed in 135. The sideways wedge plus the 5 downward wedges, which are close to each other represent 15. The space in between the two 60’s and 15 represents that each of those numbers are two different consecutive powers of 60. Using the wedges and the spaces, the Babylonians wrote numbers of great scale. Not all numbers have a specific power of 60, like 2*603 + 33*601 + 20, so the Babylonians “just left a blank space in the number they were writing”.5 But this did not always work because a space also represented the separation of two different powers of 60. If there was any context on the tablet, that made deciphering the number slightly easier, but what happened when there was no context? For a while, the Babylonians had no concept of “zero”. “At some 3 Zara, Tom. "A Brief Study of Some Aspects of Babylonian Mathematics." 4 Lombardi, Michael A. "Why Is a Minute Divided into 60 Seconds, an Hour into 60 Minutes, Yet There Are Only 24 Hours in a Day?" 5 Troutman, Jeremy. "Number Systems." Manasiya 3 point, prior to the arrival of the Seleucid Turks in 311 B.C.E, Babylonian astronomers and mathematicians devised a true zero, to indicate the absence of units of a given order of magnitude”. 6 This symbol looked like this . If we go back to the example stated earlier in this paragraph, that is 2*603 + 33*601 + 20, we would have the following cuneiform: . The first two downward wedges represent 2*603. The special zero symbol shows there is no value of 602. The three side wedges and three downward wedges represent 33*601, and the last two side wedges represent 20*600. The Babylonian numeral system wasn’t just restricted to whole numbers; writing cuneiform for fractions was also possible. To write a fraction with our current Hindu-Arabic numbers, powers of 10 are used for the denominator, but for the Babylonians, powers of 60 were 1 2 5 used in the denominator. For example, to write 0.125 in base 10 is + + . But to write 10 102 103 7 30 0.125, which is 1/8, in base 60 is + . Before the discovery of the “zero” symbol, writing 60 602 fractions was difficult because identifying which part of the cuneiform was a fraction and which part was a whole wasn’t easy to identify. But with the “zero” symbol, to write a proper fraction, just place at the beginning to indicate this number is a proper fraction. The Babylonians inherited the base 60 concept from the Sumerians, but it is not clear why base 60 was chosen over any other base. Despite not knowing the purpose behind the use of base 60 for Babylonian numerals, base 60 was used in Babylonian society and influenced today’s society as well. “The Babylonians made astronomical calculations in the sexagesimal (base 60) system”7. Today, although base 60 is not used anymore to do computations, “the sexagesimal 6 Ifrah, Georges. The Universal History of Numbers: From Prehistory to the Invention of the Computer. 7 Lombardi, Michael A. "Why Is a Minute Divided into 60 Seconds, an Hour into 60 Minutes, Yet There Are Only 24 Hours in a Day?" Manasiya 4 system is still used to measure angles, geographic coordinates and time. In fact, both the circular face of a clock and the sphere of a globe owe their divisions to a 4,000-year-old numeric system of the Babylonians.”8 The Babylonians developed a positional numeral system using both base 10 and base 60. They also, later in their reign, create a symbol for “zero”, and influenced many modern mathematical ideas we use today. Although there may be faults in the Babylonian numeral system, their work allowed the advancement of mathematics, led us to where we are today, and will take us further in mathematics than we are currently have the knowledge of. 8 Lombardi, Michael A. "Why Is a Minute Divided into 60 Seconds, an Hour into 60 Minutes, Yet There Are Only 24 Hours in a Day?" Manasiya 5 Works Cited Ifrah, Georges. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Trans. David Bellos, E. F. Harding, Sophie Wood, and Ian Monk. New York: Wiley, 2000. Print. Ji, Shanyu. History of Mathematics. Lombardi, Michael A. "Why Is a Minute Divided into 60 Seconds, an Hour into 60 Minutes, Yet There Are Only 24 Hours in a Day?" Scientific American. Scientific American, a Division of Nature America, Inc., 5 Mar. 2007. Web. 15 Sept. 2018. O'Conner, J. J., and E. F. Robertson. "Babylonian Numerals." Newton Biography. N.p., Dec. 2000. Web. 15 Sept. 2018. Troutman, Jeremy. "Number Systems." Cardano. N.p., n.d. Web. 15 Sept. 2018. Zara, Tom. "A Brief Study of Some Aspects of Babylonian Mathematics." Thesis. Liberty University, 2008. .